To what degree can a potential outbreak of Smallpox virus be modeled for Hickory High School, North Carolina, United States? Kristina StuckeyIntroduction Up until 1980, the world lived in fear of a deadly, disfiguring disease.
Smallpox, a highly infectious disease, produces symptoms such as fever, postulation, and rash, potentially leaving the victim scarred from scratching his or her blisters. It is transmitted through inhalation of fluid droplets produced by actions such as sneezing and coughing as well as contact with a skin legion of an infected person (Funk and Wagnalls). The significance of the death toll of Smallpox is hard to fathom because the disease is suspected to have been around since 30 A.D. but is estimated to be in the millions. Luckily, Edward Jenner created an effective vaccine from cowpox in the late 18th century, while evolutions in the vaccine would be continued (Belongia).
By producing an easy use vaccine, eradication efforts were made to rid the world of Smallpox. The campaign started in 1958 to completely eradicate the globe of the disease, yet were not completed until the final case was declared in 1979 after much scientific and political effort. Now the potential for Smallpox to become a biological weapon is a leering threat (Funk and Wagnalls), highlighting the importance of technology and mathematics to explore potential outbreaks and their effects on society.
If a bioterrorist attack were to occur, the effects would be detrimental to a suburban or urban community. In the United States, the government stopped providing the Smallpox vaccine in 1972. This means that anyone under the age of 44, except for certain military and government employees, would have no immunity to the virus.
There is potential for a full spread outbreak of the virus because there is no natural defense. Diseases have always fascinated me since I was a young age. I remember exploring the Center for Disease Control and Prevention’s website, intrigued by the science behind the diseases. The ability for a disease to pervade a population is stunning, moving in an invisible and silent manner. The sheer amount of destruction that can be caused by one cell or capsule is astounding, creating an increased need to understand the science. The more information scientists are able to find, the better equipped to handle disaster society will be. If a hypothetical outbreak were to occur at my hometown, our high school students would have no natural immunity.
Thus, if an outbreak were to occur, it would ravage the student body. This paper will be using the SIR model, which involves using three different differential formulas for susceptible, infected, and recovered individuals, hence the abbreviation SIR. FormulasIn this investigation, these formulas will be graphed using the Runge-Kutta algorithm and graphed by Java Script. Understanding that the sum of all three formulas must equal one because there is one population is key. The differential equations are vital in converting the numbers to decimals to add up to one. 1=S+I+RThe Susceptible Equation: dSdt=-b s(t)I(t)The Recovered Equation: drdt=k i(t)The Infected Equation: didt=b s(t) i(t)- k i(t)ConstantsD: Time of Infectious capability- For smallpox the average is 11.5 days, but for the sake of this experiment 12 days was used (Meltzer, Martin I., et a).
Ro: Basic Reproductive Rate: Typically 5-7, but 5 was used for in this experiment (University of Michigan).How to Find Variable, vDuration of Time Until one is no longer infectious (D)=1v12?1vv=112Steps to Find ?Use the formula: Ro=?v5=?112?=5×112?=512Situation 1:VariablesS=S(t) the number of individuals susceptible: 1,018I=I(t) the number of individuals infected: 24R=R(t) number of individuals recovered: 0The first situation mimics an outbreak if the Senior IB diploma students were all infected with Smallpox. Because the student body is all under the age of 44, for the sake of this paper it is assumed that no one has natural immunity. This being said, for every person that is infected, they will spread that to three other people.Graph 1: The different phases of a hypothetical Smallpox outbreak at local high school caused by an infected IB Senior class.Analysis of Graph 1The initial population is almost solely in the susceptible pool at the beginning of the graph, with a small number infected. The infected formula grows exponentially until it reaches its maximum, at which the y-values switch from increasing to decreasing. The susceptible model has a point of inflection approximately near the value of 10 for x, where the graph goes from a negative concavity to a positive.
The recovered model reflects the susceptible, with a positive concavity that changes at the point of inflection to a negative concavity. Situation 2: VariablesS=S(t) the number of individuals susceptible: 1,018I=I(t) the number of individuals infected: 255R=R(t) number of individuals recovered: 0The use of the entire senior class increases the infected population significantly, changing the shape of the curves. There are 255 members of our school’s senior class, which are all under 44, hence the assumed lack of immunity. Graph 2: The different phases of a hypothetical Smallpox outbreak at local high school caused by an infected IB Senior class.Analysis of Graph 2The infection model reaches its maximum closer to the zero than the previous graph, meaning the peak of the epidemic is completed sooner. For susceptible formula, the slope is a negative number further from zero, which can be inferred through the change demonstrated in slope. The same can be said about the recovered model, because of the greater speed the population became healthy once more.Conclusion The data was limited in validity due to inability to access public health records of those who are vaccinated out of a given population.
Issues that are to blame for the lack of available records are improper or outdated methods of recording and doctor-patient confidentiality rules. If I were a government official in charge of fixing an epidemiologic outbreak, I would have necessary tools to complete my job, unlike what I had for this experiment. This date provides a glimpse into two of the many potential outbreaks that could occur not only at our school, but in our world as well.
The data showed that when the population of infected is larger in the beginning, the diseases would work its way quicker throughout a population. To continue researching, I would investigate the influence of adding vaccination availability to a population to see how that shapes the models differently. Works CitedBelongia, Edward A., and Allison L. Naleway. “Smallpox Vaccine: The Good, the Bad, and the Ugly.” Clinical Medicine and Research, Marshfield Clinic Research Foundation, Apr.
nih.gov/pmc/articles/PMC1069029/. Accessed 11 Sept. 2017.Meltzer, Martin I., et al.
“Modeling potential responses to smallpox as a bioterrorist weapon. (Research).” Emerging Infectious Diseases, vol. 7, no. 6, 2001, p.
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” Funk & Wagnalls New World Encyclopedia, 2016, p. 1p. 1.
“The SIR Model for Spread of Disease – The Differential Equation Model.” Mathematical Association of America, Mathematical Association of America, www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-the-differential-equation-model.
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